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Let 𝕄 be a compact C∞-smooth Riemannian manifold of dimension n, n ≥ 3, and let 𝜑λ : ΔM𝜑λ + λ𝜑λ denote the Laplace eigenfunction on 𝕄 corresponding to the eigenvalue λ. We show that Hⁿ⁻¹({𝜑λ = 0}) ≤ Cλα, where α > 1/2 is a constant, which depends on n only, and C > 0 depends on 𝕄. This result is a consequence of our study of zero sets of harmonic functions on C∞-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

Let u be a harmonic function in the unit ball B(0,1) ⊂ ℝⁿ, n ≥ 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hⁿ⁻¹({u = 0} ⋂ B) ≥ c. We prove Nadirashvili's conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction 𝜑λ on M, which corresponds to the eigenvalue λ, the following inequality holds: $\\mathrm{c}\\sqrt{\\mathrm{\\lambda }}\\le {\\mathrm{H}}^{\\mathrm{n}-1}\\left(\\{{\\mathrm{\\phi }}_{\\mathrm{\\lambda }}=0\\}\\right)$.

This review focuses on the analysis of temporal beta diversity, which is the variation in community composition along time in a study area. Temporal beta diversity is measured by the variance of the multivariate community composition time series and that variance can be partitioned using appropriate statistical methods. Some of these methods are classical, such as simple or canonical ordination, whereas others are recent, including the methods of temporal eigenfunction analysis developed for multiscale exploration (i.e. addressing several scales of variation) of univariate or multivariate response data, reviewed, to our knowledge for the first time in this review. These methods are illustrated with ecological data from 13 years of benthic surveys in Chesapeake Bay, USA. The following methods are applied to the Chesapeake data: distance-based Moran's eigenvector maps, asymmetric eigenvector maps, scalogram, variation partitioning, multivariate correlogram, multivariate regression tree, and two-way MANOVA to study temporal and space–time variability. Local (temporal) contributions to beta diversity (LCBD indices) are computed and analysed graphically and by regression against environmental variables, and the role of species in determining the LCBD values is analysed by correlation analysis. A tutorial detailing the analyses in the R language is provided in an appendix.

It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

We introduce a refinement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m − pn) < ∞ for every integer m. We also show that lim inf(pn+1 − pn) ≤ 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 − pn) ≤ 12 and lim infn(pn+2 − pn) ≤ 600.

Rayleigh waves are ubiquitously used for subsurface characterization through dispersion curve inversion, whose quality depends on the number of useable overtones. Traditional analysis is based on vertical receivers and, for active surveys, sources. However, for layered media, eigenfunction theory shows that optimal recovery of any given dispersion mode and frequency can be achieved by either vertical source‐vertical receiver or radial source‐radial receiver configurations, with source directivity being more dominant. Multi‐directional near‐surface wave‐equation modeling and field examples, including distributed acoustic sensing data, validate these predictions. Statistical analysis of compaction‐type near‐surface models shows that overtones are better recovered for radial‐radial surveys in large portions of the useable spectrum, extending beyond 60% for the second and above modes. We conclude that incorporating radial‐radial data acquisition is beneficial and should become standard procedure in active surveying, as well as analyzing the radial component in ambient noise and earthquake‐induced Rayleigh waves. Plain Language Summary Surface waves are a type of seismic wave that are often utilized for near‐surface characterization in a wide variety of fields. However, the quality of their recording is highly variable between different sites. By utilizing sources and receivers that generate and record energy in different directions, we can better excite these surface waves and facilitate their subsequent analysis. We show the benefits of directional surveys through mathematical, simulative and field data examples, and claim that they should become a widespread procedure in all fields relying on surface‐wave surveys. Key Points Radial and vertical directivity of sources and receivers strongly influences excitation of different Rayleigh‐wave dispersion modes Eigenfunction theory predicts the effect of survey directivity on recorded Rayleigh waves, also critical for distributed acoustic sensing Statistical analysis of near‐surface models shows significant improvement of overtone recovery by adding radial‐radial acquisition

Electro-osmotic consolidation is taken as a valid and practical approach for soft ground improvement, and it is of vital significance to study the consolidation behaviour of electro-osmotic consolidation for the application purpose. This paper presents an analytical solution for two-dimensional (2D) electro-osmosis-enhanced preloading consolidation of unsaturated soil. The governing equations are first expanded using the eigenfunction expansion method in the y-direction, and then, the auxiliary function is constructed to rearrange the boundary conditions for mathematical convenience. The coupled partial differential equations are subsequently transformed into a system of ordinary differential equations (ODEs) containing only time t by adopting the eigenfunction expansion method in the x-direction. The exact solution is finally found by using the Laplace transform techniques to solve the ODEs system. Working example conducts a series of numerical simulations to validate the newly proposed solutions, and verification results demonstrate excellent agreement. Using the proposed solutions, this paper conducts parametric analyses to study the influences of coupled electro-osmosis and preloading on the consolidation behaviour.

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.